Global existence of solutions to a nonlocal equation with degenerate anisotropic diffusion

TL;DR

This paper proves the global existence of very weak solutions for a complex nonlocal diffusion-advection-reaction equation with degenerate anisotropic diffusion, extending previous mass-conserving models in higher dimensions.

Contribution

It introduces conditions for the admissible degeneracy of the diffusion tensor based on the upper box fractal dimension, expanding the understanding of such equations.

Findings

Global existence of solutions established

Degeneracy characterized via fractal dimension

Extension of previous models to higher dimensions

Abstract

Global existence of very weak solutions to a non-local diffusion-advection-reaction equation is established under no-flux boundary conditions in higher dimensions. The equation features degenerate myopic diffusion and nonlocal adhesion and is an extension of a mass-conserving model recently derived in arXiv:2308.05676. The admissible degeneracy of the diffusion tensor is characterised in terms of the upper box fractal dimension.